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Set theory was for a time a controversial area of mathematics because of Cantor's results and some mathematicians like Poincaré and Kronecker reacting violently to the implications. There was a small, ...
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The usual way to develop the foundations of the number systems is to define the naturals axiomatically, usually by the Peano axioms, and then to produce the rest of the numbers by rather artificial ...
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In constructing the natural numbers, are we allowed to use whatever we want as a primitive notion? I ask because of the following: Suppose the real numbers are primitive notions. Then define the ...
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Most here have probably heard the statements: “one true arithmetic” and “God gave us the integers” (Kronecker). I’m wondering if corresponding objects from two different mathematical foundations are ...
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By "all of math" I mean geometry, groups, algebra, functions, etc; that which we don't do in set theory proper even though in principle we could. How do we know that (almost) all of math ...
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We speak of numbers, sets, functions, and spaces with the same grammatical confidence as we do of trees, atoms, or mountains. Yet, unlike physical entities, mathematical objects are abstract, non-...
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In category theory, Set is just a category with sets as objects. It captures set theory in its entirety and makes it much simpler while retaining it's essential features. Has set theory been ...
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Is the following sentence a contradiction in terms in the standard interpretation of FOL given the existential import of the word “some”: Some sets don’t exist? I ask because this sentence doesn’t ...
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Let's suppose that there are two kinds of sets: those that are self-elements and those that aren't. Why would we have any reason to hope that determining which of those two kinds of sets we are ...
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I am reading Hellman’s Mathematics Without Numbers. I’ve read exactly up to page 26, and my understanding is roughly this: Hellman does not understand mathematical sentences as sentences about certain ...
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If you are into math I'm sure you have come across NJ Wildberger's provocative videos on youtube. According to this guy, the infinite has no place in mathematics. Real numbers, which are the ...
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Why did Voevodsky choose type theory as a foundation for mathematics? I understand we now have a notion of "homotopy coherent diagram" as well as opetopic sets. This is truly a big ...
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Category theory can be seen as foundational theory of mathematics since it brings together different subdomains and gives a more abstract and general framework to "ground" those subdomains. ...
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Good afternoon, I hope you are having a lovely day. I am a student of mathematics and logic and plan to specialize into philosophy of mathematics. I already know quite a bit of background on formal ...
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The more I have been deeply pondering the foundation of mathematics the more it seems like the root of all evil and ambiguity comes from the (seemingly harmless) Power Set Axiom. I'm curious as to the ...
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I always thought of mathematics as being founded on logic. After all, even the most basic mathematical definition is based on logic. When we enunciate ZFC axioms, we're relying on the concepts of &...
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I am currently exploring defining mathematical theories in terms of mental concepts rather than as being formalized in yet other mathematical theories. In the single-sorted presentation of NBG class ...
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I have some questions related to multivalued logic. I am new to this forum, (I study mathematics) so I would be grateful to any useful advice. I am doubtful on even posting this question on Philosophy ...
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Vladimir Arnold (b. 1937) once said that David Hilbert (b. 1862) and Bourbaki (f. 1934) "proclaimed that the goal of their science was the investigation of all corollaries of arbitrary systems of ...
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Is there a kind of "consensus" towards the meaning & intuition of the concept of "potential infinity" that goes back to Aristotle and is promoted by Edward Nelson, e.g. in the ...
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I m reading Linnet's paper 'pluralities and set' where his claim said that collapse principle lead contradiction if we didn't assume 'it is possible to quantify over absolutely everything' He uses ...
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Per Hamkins[[11][12]] (see also his [22]), if no individual axiom is too sacred to be denied in some possible world,Q and so if no collection of such axioms is so sacred either, yet then: The ...
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So I've been on a number of math fora, part of learning some calculus (not much of set theory, no). To my surprise I found what I would describe as strong resistance from some folks against (using) ...
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I am reading George Boolos's "The Iterative Concept of Set," and in the first chapter of this paper, he criticizes Cantor's definition of a set as a whole or totality of objects, pointing ...
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Here are the main components to my understanding on this issue: Almost all of math can be given in a foundation of set theory Different math can seemingly contradict, e.g. in Euclidean geometry ...
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I get infinite regress but surely the axioms of ZFC or arithmetic were not so much chosen as discovered and intuited and thought about. They certainly didn't just grab whatever was around them and say ...
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I find the common presentation of first order logic somewhat confusing. I feel that I often don’t understand why we need the exact terms and concepts we do. My current recapitulation of “standard FOL” ...
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sometimes one sees/reads assertions such as "[the bounded inverse theorem] is equivalent to both the open mapping theorem and the closed graph theorem", but taken formally and literally this ...
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A deductive system (as a collection of judgments and rules of inference) can be used to describe something commonly called a “set theory”. We can imagine a priori there are certain properties we would ...
Julius Hamilton's user avatar
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What you have cited is a pragmatic limit, as you have not seen logic systems with more than 8 or so precepts. IF there were such a limit to precept quantity, then YES there would be a limit to the ...
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The text Abstract and Concrete Categories: The Joy of Cats makes much of the class/set distinction, including as a foundational matter, and situates this distinction relative to another such notion, ...
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Is this a settled (as much as it can be) philosophical area? I feel like I understand that there will always be incompleteness for a finite set of axioms trying to capture all of arithmetic. But I ...
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It’s been on my mind lately. I do maths and work with them daily, but I’m not entirely sure of what they really are. I understand they are symbols at a surface level, but there is obviously more to it....
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On first principles, Wikipedia says: A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of ...
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I thought I had a grasp on this. Do Gödel's apply to just math; logic, too; or more, and what does its applicability entail? If it applies to math, does it apply to physics? Similarly with Tarski: can ...
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This question is related to another question I posted but I think it requires its own treatment since of the already wide scope of the other question i.e. Is the classical theory of concepts ...
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Math is obviously based on logic in a heirarchical sense, but what about the historical sense? Is there any historical evidence of a "transition" from first order logic to mathematics? All ...
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I'm not asking whether mathematics is discovered or invented, rather whether being theist implies/strengthens/related to the view that it is discovered, and vice versa. For example I came across an ...
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Let's go back to the original meanings of addition and multiplication back in ancient Sumer when arithmetic was primarily used as a tool in the trade of sheep and beer. Addition meant something like ...
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I have thought a lot a about mathematics and it's foundations. There have been several attempts to give it a solid foundation, and they all failed. Frege / Russell logical atomist approach failed, ...
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Humans have been doing mathematics for at least 3000 years. In ancient Egypt they did some advanced trigonometry and number theory. Mathematics is thousands of years old, but for some reason it was ...
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Frege's ill-fated program to define the natural numbers in terms of abstraction was intended as a genuine account of what natural numbers are, not just a way of encoding numbers in set theory like ...
David Gudeman's user avatar
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Some definitions (from what I can tell): A multigraph is a graph where a node can connect via multiple edges. A hypergraph is a graph where a single edge can connect more than two nodes. ...
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How would we define a topos between Classical Logic and Paraconsistent Logic, and determine what the homomorphism between them would be? I learned that topoi can't be used for philosophical ideas, but ...
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Is the cogito an axiom from which we can reason axioms of mathematics? Was Descartes' aim to make mathematics (and other fields of knowledge) reducible to the cogito?
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Wiles' proof initially involved reference to functional equivalents of inaccessible cardinals (here, Grothendieck universes). Rather than take this as evidence for the meaningfulness and usefulness of ...
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After reading this essay about the history of type theory, I have refined my assessment of the set- vs. type-theory question in two ways. More similarly to what I was thinking before, I still ground ...
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All mathematicians are familiar with the (extremely plausible) fact that any ordinary mathematical proof can be formalized inside some foundational theory, e.g., ZFC. Why doesn't this imply that ...
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Do philosophers distinguish between mathematical creativity, and the broader artistic creativity? If so, what are the differences between these two? A lot of people seem to treat IQ as something ...
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Does every mathematical question have an unambiguous answer? For example, suppose I were to assert "In the decimal expansion of pi, does there occur in at least one location a billion 1's in a ...
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